Tighter bounds of errors of numerical roots

Abstract

LetP(z) be a monic univariate polynomial overC, of degreen and having roots ζ1,..., ζn. Given approximate rootsz1,...,zn, with ζi ≃zi (i = 1,...,n), we derive a very tight upper bound of ¦ζi −zi¦, by assuming that ζi has no close root. The bound formula has a similarity with Smale’s and Smith’s formulas. We also derive a lower bound of ¦ζi −zi¦ and a lower bound of min{ζj −zi¦ ¦j ≠i}.

Keywords

Error Bound Tight Bound Numerical Root Approximate Root Close Root

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Work supported in part by Japanese Ministry of Education, Science and Culture under Grants 15300002.